11n
49
(K11n
49
)
A knot diagram
1
Linearized knot diagam
6 1 7 9 2 10 1 11 6 4 8
Solving Sequence
1,7 8,10
6 2 3 4 5 9 11
c
7
c
6
c
1
c
2
c
3
c
5
c
9
c
11
c
4
, c
8
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
5
+ u
4
− 4u
3
+ 6b − 6u − 2, u
3
− 2u
2
+ 4a + 4u − 2, u
6
− 2u
5
+ 8u
4
− 4u
3
+ 12u
2
+ 8u + 4i
I
u
2
= hb + 1, 2a
2
+ au + 4a + u + 1, u
2
+ 2i
I
v
1
= ha, b − 1, v
2
− v + 1i
* 3 irreducible components of dim
C
= 0, with total 12 representations.
1
The image of knot diagram is generated by the software “Draw programme” developed by An-
drew Bartholomew(http://www.layer8.co.uk/maths/draw/index.htm#Running-draw), where we modi-
fied some parts for our purpose(https://github.com/CATsTAILs/LinksPainter).
2
All coefficients of polynomials are rational numbers. But the coefficients are sometimes approximated
in decimal forms when there is not enough margin.
1
I. I
u
1
= h−u
5
+ u
4
− 4u
3
+ 6b − 6u − 2, u
3
− 2u
2
+ 4a + 4u − 2, u
6
− 2u
5
+
8u
4
− 4u
3
+ 12u
2
+ 8u + 4i
(i) Arc colorings
a
1
=
0
u
a
7
=
1
0
a
8
=
1
−u
2
a
10
=
−
1
4
u
3
+
1
2
u
2
− u +
1
2
1
6
u
5
−
1
6
u
4
+
2
3
u
3
+ u +
1
3
a
6
=
−
1
12
u
5
+
1
3
u
4
+ ··· + u
2
+
5
6
−
1
3
u
5
+
1
3
u
4
−
4
3
u
3
+
1
3
a
2
=
1
6
u
5
−
2
3
u
4
+
11
12
u
3
− u
2
−
1
6
u
5
−
7
4
u
4
+ 3u
3
− 2u
2
−
1
2
u − 1
a
3
=
1
6
u
5
−
2
3
u
4
+
11
12
u
3
− u
2
−
1
6
−
1
12
u
5
+
7
12
u
4
+ ··· +
3
2
u +
1
3
a
4
=
1
12
u
5
−
1
12
u
4
+ ··· +
3
2
u +
1
6
−
1
12
u
5
+
7
12
u
4
+ ··· +
3
2
u +
1
3
a
5
=
1
3
u
5
+
1
6
u
4
+
13
12
u
3
− u +
1
6
−0.416667u
5
+ 5.41667u
4
+ ··· + 8.50000u + 3.66667
a
9
=
u
2
+ 1
−u
4
− 2u
2
a
11
=
−u
u
3
+ u
a
11
=
−u
u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = u
5
− 2u
4
+ 8u
3
− 5u
2
+ 12u + 4
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
6
− 7u
5
+ 30u
4
− 59u
3
+ 78u
2
− 23u + 9
c
2
u
6
+ 11u
5
+ 230u
4
+ 895u
3
+ 3910u
2
+ 875u + 81
c
3
u
6
− u
5
+ 4u
4
+ 203u
3
+ 402u
2
− 199u + 127
c
4
u
6
− 13u
5
+ 64u
4
− 127u
3
+ 74u
2
+ 17u + 41
c
6
, c
9
u
6
+ 4u
5
+ 9u
4
+ 8u
3
+ 19u
2
+ 4u + 3
c
7
, c
8
, c
11
u
6
+ 2u
5
+ 8u
4
+ 4u
3
+ 12u
2
− 8u + 4
c
10
u
6
+ u
5
+ 4u
4
+ u
3
+ 8u
2
+ 5u + 3
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
6
+ 11y
5
+ 230y
4
+ 895y
3
+ 3910y
2
+ 875y + 81
c
2
y
6
+ 339y
5
+ ··· − 132205y + 6561
c
3
y
6
+ 7y
5
+ 1226y
4
− 38137y
3
+ 243414y
2
+ 62507y + 16129
c
4
y
6
− 41y
5
+ 942y
4
− 6133y
3
+ 15042y
2
+ 5779y + 1681
c
6
, c
9
y
6
+ 2y
5
+ 55y
4
+ 252y
3
+ 351y
2
+ 98y + 9
c
7
, c
8
, c
11
y
6
+ 12y
5
+ 72y
4
+ 216y
3
+ 272y
2
+ 32y + 16
c
10
y
6
+ 7y
5
+ 30y
4
+ 59y
3
+ 78y
2
+ 23y + 9
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.327848 + 0.380167I
a = 0.782599 − 0.521714I
b = 0.089037 + 0.417324I
0.080134 − 1.031470I 1.24075 + 6.28341I
u = −0.327848 − 0.380167I
a = 0.782599 + 0.521714I
b = 0.089037 − 0.417324I
0.080134 + 1.031470I 1.24075 − 6.28341I
u = 0.31945 + 1.74021I
a = −0.565198
b = −0.20409 + 1.44525I
−4.41014 − 1.50896I −1.48189 + 1.11182I
u = 0.31945 − 1.74021I
a = −0.565198
b = −0.20409 − 1.44525I
−4.41014 + 1.50896I −1.48189 − 1.11182I
u = 1.00840 + 2.01334I
a = 0.782599 + 0.521714I
b = 2.11506 − 1.80559I
9.26481 + 6.90911I −1.75886 − 2.47219I
u = 1.00840 − 2.01334I
a = 0.782599 − 0.521714I
b = 2.11506 + 1.80559I
9.26481 − 6.90911I −1.75886 + 2.47219I
5
II. I
u
2
= hb + 1, 2a
2
+ au + 4a + u + 1, u
2
+ 2i
(i) Arc colorings
a
1
=
0
u
a
7
=
1
0
a
8
=
1
2
a
10
=
a
−1
a
6
=
a + 1
−1
a
2
=
a +
1
2
u + 1
−au
a
3
=
a +
1
2
u + 1
−au − 2a − u − 2
a
4
=
−au − a −
1
2
u − 1
−au − 2a − u − 2
a
5
=
a +
1
2
u + 1
−au − 2a − u − 2
a
9
=
−1
0
a
11
=
−u
−u
a
11
=
−u
−u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4au − 4u − 4
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
10
(u
2
+ u + 1)
2
c
3
u
4
− 2u
3
+ u
2
− 6u + 9
c
4
u
4
+ 2u
3
+ u
2
+ 6u + 9
c
5
(u
2
− u + 1)
2
c
6
(u + 1)
4
c
7
, c
8
, c
11
(u
2
+ 2)
2
c
9
(u − 1)
4
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
10
(y
2
+ y + 1)
2
c
3
, c
4
y
4
− 2y
3
− 5y
2
− 18y + 81
c
6
, c
9
(y −1)
4
c
7
, c
8
, c
11
(y + 2)
4
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.414210I
a = −0.387628 − 0.353553I
b = −1.00000
−6.57974 + 2.02988I −6.00000 − 3.46410I
u = 1.414210I
a = −1.61237 − 0.35355I
b = −1.00000
−6.57974 − 2.02988I −6.00000 + 3.46410I
u = − 1.414210I
a = −0.387628 + 0.353553I
b = −1.00000
−6.57974 − 2.02988I −6.00000 + 3.46410I
u = − 1.414210I
a = −1.61237 + 0.35355I
b = −1.00000
−6.57974 + 2.02988I −6.00000 − 3.46410I
9
III. I
v
1
= ha, b − 1, v
2
− v + 1i
(i) Arc colorings
a
1
=
v
0
a
7
=
1
0
a
8
=
1
0
a
10
=
0
1
a
6
=
1
−1
a
2
=
2v
−v
a
3
=
2v − 1
−v
a
4
=
v − 1
−v
a
5
=
2v − 1
−v
a
9
=
1
0
a
11
=
v
0
a
11
=
v
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4v + 2
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
u
2
+ u + 1
c
3
, c
4
, c
5
c
10
u
2
− u + 1
c
6
(u − 1)
2
c
7
, c
8
, c
11
u
2
c
9
(u + 1)
2
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
10
y
2
+ y + 1
c
6
, c
9
(y − 1)
2
c
7
, c
8
, c
11
y
2
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = 0.500000 + 0.866025I
a = 0
b = 1.00000
−1.64493 + 2.02988I 0. − 3.46410I
v = 0.500000 − 0.866025I
a = 0
b = 1.00000
−1.64493 − 2.02988I 0. + 3.46410I
13
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
2
+ u + 1)
3
(u
6
− 7u
5
+ 30u
4
− 59u
3
+ 78u
2
− 23u + 9)
c
2
(u
2
+ u + 1)
3
(u
6
+ 11u
5
+ 230u
4
+ 895u
3
+ 3910u
2
+ 875u + 81)
c
3
(u
2
− u + 1)(u
4
− 2u
3
+ u
2
− 6u + 9)
· (u
6
− u
5
+ 4u
4
+ 203u
3
+ 402u
2
− 199u + 127)
c
4
(u
2
− u + 1)(u
4
+ 2u
3
+ u
2
+ 6u + 9)
· (u
6
− 13u
5
+ 64u
4
− 127u
3
+ 74u
2
+ 17u + 41)
c
5
(u
2
− u + 1)
3
(u
6
− 7u
5
+ 30u
4
− 59u
3
+ 78u
2
− 23u + 9)
c
6
(u − 1)
2
(u + 1)
4
(u
6
+ 4u
5
+ 9u
4
+ 8u
3
+ 19u
2
+ 4u + 3)
c
7
, c
8
, c
11
u
2
(u
2
+ 2)
2
(u
6
+ 2u
5
+ 8u
4
+ 4u
3
+ 12u
2
− 8u + 4)
c
9
(u − 1)
4
(u + 1)
2
(u
6
+ 4u
5
+ 9u
4
+ 8u
3
+ 19u
2
+ 4u + 3)
c
10
(u
2
− u + 1)(u
2
+ u + 1)
2
(u
6
+ u
5
+ 4u
4
+ u
3
+ 8u
2
+ 5u + 3)
14
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
(y
2
+ y + 1)
3
(y
6
+ 11y
5
+ 230y
4
+ 895y
3
+ 3910y
2
+ 875y + 81)
c
2
((y
2
+ y + 1)
3
)(y
6
+ 339y
5
+ ··· − 132205y + 6561)
c
3
(y
2
+ y + 1)(y
4
− 2y
3
− 5y
2
− 18y + 81)
· (y
6
+ 7y
5
+ 1226y
4
− 38137y
3
+ 243414y
2
+ 62507y + 16129)
c
4
(y
2
+ y + 1)(y
4
− 2y
3
− 5y
2
− 18y + 81)
· (y
6
− 41y
5
+ 942y
4
− 6133y
3
+ 15042y
2
+ 5779y + 1681)
c
6
, c
9
(y − 1)
6
(y
6
+ 2y
5
+ 55y
4
+ 252y
3
+ 351y
2
+ 98y + 9)
c
7
, c
8
, c
11
y
2
(y + 2)
4
(y
6
+ 12y
5
+ 72y
4
+ 216y
3
+ 272y
2
+ 32y + 16)
c
10
(y
2
+ y + 1)
3
(y
6
+ 7y
5
+ 30y
4
+ 59y
3
+ 78y
2
+ 23y + 9)
15
